Abstract This web page currently devotes itself fully to Conway's game of Life. In the long run however its aim is eventually to encompass a wider realm of topics including Cellular Automaton and Nature Modeling. One won't find much stuff available here; by some time, Life was to me a pretty enjoyable topic of study with plenty of time for it. Life is not presently one of my priorities but I keep it in mind and proceeds working on a few projects you shall find available sooner or later. For that matter, it is easier to start a web page than to fulfill its visitor's expectations. This is at the moment far from being comprehensive and rigorous. If you are likely to be offended by such lack of scientific integrity, you are kindly requested to leave and come back later.

Life is a cellular automaton on an infinite square lattice with semi-totalistic rules, designed by mathematician John Horton Conway in the 70's. In less specialized terms, Life can naively be described in term of "cells" susceptibles of two states, named Alive and Dead, on an unbounded chess-like board, with rules to be applied iteratively and simultaneously to each cell according to the number of alive cells surrounding it. Those rules endow the "Life Universe" with dramatics and worth studying behaviors. There exists many equivalent definitions, with no real best exposure despite most exotic attempts; the following is as Conway stated them in one of his rare publication on the subject [Berlekamp1]:

BIRTH. A cell that's dead at time t becomes live at t+1 only if exactly three of its eight neighbors were live at t.

DEATH by overcrowding. A cell that's live at t and has four or more of its eight neighbors live at t will be dead by time t+1.

DEATH by exposure. A live cell that has only one live neighbor, or none at all, at time t, will also be dead at t+1.

SURVIVAL. A cell that was live at time t will remain live at t+1 if and only if it had just 2 or 3 live neighbors at time t.

The invaluable discovery that those laws can be reduced to three (or less!) is of no interest, whatever aspect you feel concerned with. That does not simplify in any way Conway's rules, neither it does clarify the statement, intended for human being or as a prelude of an algorithm. The best definition lies in mathematics, which we shall not discuss here. Conway's presentation has advantage of emphasizing the historical way the game came to life, for its author searching for some self-dependent system imitating living system's behavior (hence the name), lying in an average crawling between rapid decay and unlimited growth. Conway's discovery is much satisfactory in all regards. One can prove by clever fitting of life structures (to be discussed later on) Life to be a Turing machine, or universal computer, i.e., a machine capable of handling and solving any computational problem (if given the time to solve it). This is also making Life a chaotic system with no general scheme for forecasting fate of any configuration. Life has been studied by many peoples after it was presented by Martin Gardner in his famous Scientific American chronicle. Famous theoricists and explorers of life are S. Golomb, D. Bell, B. Gosper, ... (to be discussed later on). The animation above contains many characteristic life's behaviors. To ease study of Life, a whole formalism has been designed. For instance, all frequent forms or interesting ones have been given a name, and so does families of forms sharing likewise behaviors. Each little black square is one of the so-called alive cell. Four of them tightly stuck to make a bigger box, called a block, belongs to the family of still life, for their structure remains the same as the games evolves (the name structure is not standard, but we shall find useful this terminology). One can see a more complicated life form satisfying the still life's definition, namely the set of seven cells lying on top, all together called a fish hook in reference of its shape. The careful reader might have paid attention that those forms are still only part of the time, for an attentive inspection will show the structures are altered before recovering their immutable aspect a bit later on. This is a very important reaction of a certain family of Life forms to some kind of "attack". This family is called the class of eaters. Their use in controling Life processes is obvious on the basis of the animation. Equally interesting patterns are those traveling in the space. There is much to be told about what is heard by traveling here (to be discussed later on), yet an intuitive understanding will suffice for the moment. Moving patterns are called ships. Ships traveling in diagonal are called gliders and certainly are the most famous structures in Life. Bigger ships traveling downwards are a special flavor of fishes, this one identified as the LWSS. There eventually occurs a much terrific reaction by whose process a glider turns into a fish. We shall have a very great deal with this Life's aptitude later on.

Life exhibits many quantum particles' behavior, even though it does more as a cellular automaton than as the special case coined by Conway and inherited with much refinements by a whole generation of so-called Lifenthusiasts. From now on the discussion will be unduly informal and imprecise. The reason is that the basis of what follows has still no correct mathematical background for I shamefully did not succeed in doing so. Basic idea is to equip the space with a probability and providing some statistical behaviors with chaotical discrepancy. The fact that Life is unpredictable does not (compulsory) means we cannot infer anything from what has happened and how it happened. Mathematical probabilities tends to disregard any experimental issue in its intimate functioning, and supplying a probability for such a space does not matter. Yet it has to offer some relationship to Life events, which enters in the context of a very classical experimental science. There are many shocking analogies between Life establishing the complexity and "real world"'s particles making batch of matter. One can see Conway's rules being what boson are to physics, leading the "how to do", and the cells what fermions are to physics, objects constituting structures (matter to pursue the analogy). Schrodinger's individuality of a quantum particle applies as well for a Life's cell. The reaction with a line of gliders on some back and forth strolling life form with as a results a line of fishes could be put in parallel with a beam of, say, leptons, striking a target with emission of pions. The conceptually unfathomed reaction involving quantum creatures finds a straightforward easily minded analogy with cellular automata (among whose there's Life). One barely accept one particle can disintegrate when into two others, when into three others of the same type, or that particles can turn into some of other type, or likewise behaviors. Such crazy behaviors observed with quantum matter finds appealing analogy with those observed in Life. In the analogy, mass in physics has complexity as its counterpart in Life: the more complex structure you are playing Life with, the more likely your are to exhibit complex structures. For instance the glider which is a simple structure (counting only five cells; next smallest space ship, that we know for it is the LWSS, counts 9 of them) occurs systematically when pattern has a reasonnable yet small size. Gliders are easy to make (from randomness).

Demography plots are plots of the population size (number of cells) with respect to the generation. I opened this site to provide such demography of some Life Methuselahs (little patterns which offer a long and much intricate behavior before stabilizing) and my code for computing them. Demography are mainly interesting for patterns such as Methuselahs or singles but peculiar patterns (demography of a blockpusher for instance is of no use). I shall soon replace incoming plots by numerical data much less space consuming, and move graphics in another place, but as they're still here now, let's deal with them. The first one is the full R-Pentomino demography (full means till convergence, resulting in a line with possibles little periodic perturbations in the the graphics). The R-Pentomino is the first Methuselah discovered (by Conway). It indeed is small for it has only 5 cells, and stabilizes in 1103 generations. Its structure is the following one:

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The next one is acorn, one other Methuselah stabilizing in 5206 generations. I put its structure at generation 0, and two plots, one of the very start, and the whole demography before convergence.

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Here below is bunnies. This one is more interesting for it does not converge till 17332 generations. Here it is not shown till stabilization, its demography has been limited to first 7000 generations.

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There is something noticeable: demography is subjec to a 'quick growth' at its maxima of demography, just when there is the higher (local) maximum of cells, the demography is kind of exploding, then it decreases a lot. It may be related to some possibility to go beyond Life computational irreducibility (one could know what happens before computing it) at a distance of one unit, a local and temporary possibility. To illustrates this roughly, if it takes, say, one hour to compute the following configuration, then one by this mere observation can in a much lower amount of time predicts the incoming behavior (decreasing). There also are akin behavior in actual physical system, not directly related with time, but with energy, etc... (to be discussed later on).

I finally become interested in the fractal behavior of Life, where structures are exactly reproduced at a lower scale, iterating themselves. Here below is one such fractal. Each little point (pixel) is a cell. Tiny chunk one see escaping are gliders.

REFERENCES.

[Berlekamp1] Winning Ways
For your mathematical plays.
Volume 2: Games in Particular
Elwyn R. Berlekamp, John H. Conway, Richard K. Guy.
Academic Press 1985.

[Gardner1] Mathematical Games
The fantastic combinations of John Conway's new solitaire game "life"
Scientific American

[Gardner2] Mathematical Games
On cellular automata, self-reproduction, the Garden of Eden and the game "life"
Scientific American

GREETINGS AND LINKS.

Paul Callahan's web page may be is the root of most other locations for Life on the Web.

Greetings go to Peter Gerwinsky, a physicist and GNU Pascal developer. (I used GPC (GNU Pascal compiler) for my first implementation of Conway's rules)

Greetings go to Antreas P. Hatzipolakis, the maintainer of a fair comprehensive page on J.H. Conway. Antreas the Cretan also provides very interesting documents about mathematical related topics. It is where to go if you are looking for other Life sites: links are sorted and constantly updated.

I can be joined for any comments by email.